# Interquartile, Semi-Interquartile and Mid-quartile Ranges

In a set of data, the quartiles are the values that divide the data into four equal parts. The median of a set of data separates the set in half.

The median of the lower half of a set of data is the lower quartile (LQ) or ${Q}_{1}$.

The median of the upper half of a set of data is the upper quartile (UQ) or ${Q}_{3}$.

The upper and lower quartiles can be used to find another measure of variation call the interquartile range.

The interquartile range or IQR is the range of the middle half of a set of data. It is the difference between the upper quartile and the lower quartile.

Interquartile range = ${Q}_{3}-{Q}_{1}$

In the above example, the lower quartile is $52$ and the upper quartile is $58$.

The interquartile range is $58-52$ or $6$.

Data that is more than $1.5$ times the value of the interquartile range beyond the quartiles are called outliers.

Statisticians sometimes also use the terms semi-interquartile range and mid-quartile range.

The semi-interquartile range is one-half the difference between the first and third quartiles. It is half the distance needed to cover half the scores.  The semi-interquartile range is affected very little by extreme scores.  This makes it a good measure of spread for skewed distributions. It is obtained by evaluating $\frac{{Q}_{3}-{Q}_{1}}{2}$.

The mid-quartile range is the numerical value midway between the first and third quartile.  It is one-half the sum of the first and third quartiles.  It is obtained by evaluating $\frac{{Q}_{3}+{Q}_{1}}{2}$.

(The median, midrange and mid-quartile are not always the same value, although they may be.)